The complexity of many real-world problems sometimes makes it difficult or impossible to achieve a single, best objective. It may instead only be possible to identify an optimal combination of multiple, partially achieved objectives. In solving such problems, objectives are correlated to each other through one or more control variables, and the achievement level of each objective is calculated as a function of the applicable control variable values. An optimal solution is identified that provides the maximum overall achievement level as measured by some combination of individual achievement levels. Because of the complexity of the relationships that can exist between control variables and objectives, altering a control variable value to increase the achievement level of one objective may decrease the achievement level of another objective. In extreme cases, some objectives may in fact be mutually exclusive, requiring a reevaluation of what achievement levels are acceptable for each objective.
The drilling of oil & gas wells is an example of such a problem, where even just a few control variables (e.g., weight on bit, drill bit rotational speed and drilling fluid flow rate) can impact a number of differing objectives in widely varying ways. Such objectives may include, for example, maximizing the rate or penetration, keeping the equivalent circulating density below the fracture gradient, minimizing the frequency of drill bit replacement, and minimizing vibrations at the bottom-hole assembly. Thus, for example, increasing the weight on bit may increase the rate of penetration (generally desirable), but may also increase the frequency with which the drill string must be tripped to replace worn drill bits (generally undesirable).
While a number of mathematical techniques exist for performing multi-objective optimization, many of these techniques focus on finding closed-form solutions, i.e., solutions that can be expressed analytically in terms of a bounded number of well-known functions (e.g., constants, single variables, elementary arithmetic operations, nth roots, exponents, logarithms, etc.). As already alluded to, objectives may conflict to such a degree as to preclude analytically expressing the optimization problem in closed form. In such cases, recursive techniques that attempt to iteratively combine the objectives until an acceptable optimization is identified will typically diverge without identifying a solution. Further, such techniques provide little if any feedback usable to identify what steps may be taken to resolve a conflict between objectives, or to identify how variations in the control variable values affect the degree of incompatibility between the objectives.
It should be understood, however, that the specific embodiments given in the drawings and detailed description thereto do not limit the disclosure. On the contrary, they provide the foundation for one of ordinary skill to discern the alternative forms, equivalents, and modifications that are encompassed together with one or more of the given embodiments in the scope of the appended claims.